Large induced subgraphs with $k$ vertices of almost maximum degree

Abstract

In this note we prove that for every integer $k$, there exist constants $g_1(k)$ and $g_2(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at least $n-g_1(k)\sqrt{\Delta }$ vertices, such that $H$ has $k$ vertices of the same degree of order at least $\Delta (H)-g_2(k)$. This solves a conjecture of Caro and Yuster up to the constant $g_2(k)$.

arXiv comment6 pages